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Wall-crossing of D4-branes using flow trees

Jan Manschot

TL;DR

The paper extends the attractor flow-tree framework to D4–D2–D0 BPS-states with up to three endpoints, proving that edge-flow signs can be determined iteratively from the moduli and that a key indefinite quadratic form is positive definite for stable trees with N ≤ 3. This ensures the convergence of the mixed-ensemble BPS partition function in the large-volume limit and clarifies how non-primitive wall-crossing should be encoded using rational invariants bar{Omega}, consistent with S-duality. It also connects the flow-tree contributions to KS wall-crossing and introduces the role of mock Siegel theta functions and modular completions in achieving duality-consistent partition functions. Remaining challenges include constructing a complete modular completion for N=3 and extending the positivity/convergence results to higher-endpoint trees, along with a fuller incorporation of equal-charge endpoints and cross-cone walls.

Abstract

The moduli dependence of D4-branes on a Calabi-Yau manifold is studied using attractor flow trees, in the large volume limit of the Kahler cone. One of the moduli dependent existence criteria of flow trees is the positivity of the flow parameters along its edges. It is shown that the sign of the flow parameters can be determined iteratively as function of the initial moduli, without explicit calculation of the flow of the moduli in the tree. Using this result, an indefinite quadratic form, which appears in the expression for the D4-D2-D0 BPS mass in the large volume limit, is proven to be positive definite for flow trees with 3 or less endpoints. The contribution of these flow trees to the BPS partition function is therefore convergent. From non-primitive wall-crossing is deduced that the S-duality invariant partition function must be a generating function of rational, multi-covering invariants instead of integer invariants.

Wall-crossing of D4-branes using flow trees

TL;DR

The paper extends the attractor flow-tree framework to D4–D2–D0 BPS-states with up to three endpoints, proving that edge-flow signs can be determined iteratively from the moduli and that a key indefinite quadratic form is positive definite for stable trees with N ≤ 3. This ensures the convergence of the mixed-ensemble BPS partition function in the large-volume limit and clarifies how non-primitive wall-crossing should be encoded using rational invariants bar{Omega}, consistent with S-duality. It also connects the flow-tree contributions to KS wall-crossing and introduces the role of mock Siegel theta functions and modular completions in achieving duality-consistent partition functions. Remaining challenges include constructing a complete modular completion for N=3 and extending the positivity/convergence results to higher-endpoint trees, along with a fuller incorporation of equal-charge endpoints and cross-cone walls.

Abstract

The moduli dependence of D4-branes on a Calabi-Yau manifold is studied using attractor flow trees, in the large volume limit of the Kahler cone. One of the moduli dependent existence criteria of flow trees is the positivity of the flow parameters along its edges. It is shown that the sign of the flow parameters can be determined iteratively as function of the initial moduli, without explicit calculation of the flow of the moduli in the tree. Using this result, an indefinite quadratic form, which appears in the expression for the D4-D2-D0 BPS mass in the large volume limit, is proven to be positive definite for flow trees with 3 or less endpoints. The contribution of these flow trees to the BPS partition function is therefore convergent. From non-primitive wall-crossing is deduced that the S-duality invariant partition function must be a generating function of rational, multi-covering invariants instead of integer invariants.

Paper Structure

This paper contains 11 sections, 4 theorems, 83 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Wall-crossing and flow trees
  3. Kontsevich-Soibelman wall-crossing formula
  4. Supergravity and flow trees
  5. D4-D2-D0 BPS-states
  6. BPS mass and stability
  7. One endpoint
  8. Two endpoints
  9. Three endpoints
  10. Non-primitive wall-crossing
  11. Summary and discussion

Key Result

Proposition 1

Figures (3)

  • Figure 1: The attractor flow tree corresponding to $((\Gamma_1,\Gamma_2),((\Gamma_3,\Gamma_4),\Gamma_5))$.
  • Figure 2: Flow tree $T_{(12)3}$ corresponding to $(12)3$.
  • Figure 3: An indefinite lattice; the lattice points inside the green region contribute to the theta function defined in the text.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof